∫ calculus

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{x e^{\cos(10 x^2)} + \log\left(\left(\frac{1}{\tan x} + 1\right)^x\right)}{x^2 + x \sin\left(3 + \arccos(

1. **Problem statement:** (a) Given that $f'$ is continuous, $f(8) = 0$, and $f'(8) = 11$, evaluate the limit

1. The problem asks for the points of inflection of the original function $f$ and the intervals where $f$ is concave down, given the graph of its derivative $f'$. 2. Recall that po

1. **Problem statement:** We are given the graph of the derivative $f'$ and asked to find: (a) The $x$-values where the points of inflection of $f$ occur.

1. The problem is to find the intervals where a function is concave up or concave down. 2. Concavity depends on the second derivative $f''(x)$ of the function $f(x)$.

1. **State the problem:** We are given the second derivative of a function: $$f''(x) = \frac{18x^2 - 12}{(x^2 + 2)^3}$$

1. The problem asks to find the x-value where the second derivative of the function $f(x)$ changes sign from negative to positive. 2. The second derivative changing from negative t

1. The problem asks to find the x-value where the second derivative of the function $f(x)$ changes sign from negative to positive. 2. The second derivative changing sign from negat

1. The problem asks to find the x-value where the second derivative of the function $f(x)$ changes sign from negative to positive. 2. The second derivative changing sign from negat

1. The problem is to understand how to differentiate a function, which means finding its derivative. 2. Differentiation is the process of finding the rate at which a function chang

1. **Problem 1: Determine the truth of limit statements for the function $y=f(x)$ given the graph.** - The graph is piecewise linear with points: $(-1,-1)$ solid, $(0,0)$ open, $(1

1. Stating the problem: Calculate the limit \(\lim_{x \to 1} \sqrt{x^2 + 3x} - 4\). 2. Evaluate the expression inside the square root at \(x=1\): \(1^2 + 3 \times 1 = 1 + 3 = 4\).

1. **Problem statement:** Evaluate the following limits and analyze the given functions and expressions step-by-step. ---

1. **State the problem:** Find the equation of a curve passing through the point (1, 1) such that the perpendicular distance of the origin from the normal at any point $P(x,y)$ on

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = y e^x$$ with the initial condition $$x=0, y=e$$. We need to find the value of $$y$$ when $$x=1$$.

1. Problem: Find $$\lim_{x \to 2} \frac{x - 2}{4 + 2x}$$. Step 1: Substitute $x = 2$ directly.

1. **State the problem:** We want to find the value of $$\ln y = \lim_{x \to 0} \frac{\tan x - \sin x}{3x^2}$$ and then find $$y$$ itself. 2. **Recall series expansions:** For smal

1. **State the problem:** We have the function $$f(x) = \frac{1}{1-x}$$ for $$0 \leq x < 1$$ and $$f(1) = 0$$. We want to explain why $$f$$ has a minimum value but no maximum value

1. **State the problem:** Evaluate the definite integral $$\int_{-2}^3 (3x^2 - 2x - 12) \, dx$$. 2. **Find the antiderivative:** Integrate each term separately:

1. **State the problem:** Find $\frac{dy}{dx}$ for the implicit function defined by $$e^x - e^y = x - y.$$\n\n2. **Differentiate both sides with respect to $x$: **\nUsing implicit

1. **State the problem:** We need to evaluate the integral $$\int \cot^7(3\theta) \, d\theta$$. 2. **Rewrite the integral:** Recall that $$\cot x = \frac{\cos x}{\sin x}$$, so $$\c